topological crystalline insulators with a step edge*

Surfaces of multilayer semiconductors typically have regions of atomically flat terraces separated by atom-high steps. Here we investigate the properties of the low-energy states appearing at the surface atomic steps in Sn_xPb_1-xTe_ySe_1-y. We identify the important approximate symmetries and use them to construct relevant topological invariants. We calculate the dependence of mirror- and spin-resolved Chern numbers on the number of layers and show that the step states appear when these invariants are different on the two sides of the step. Moreover, we find that a particle-hole symmetry can protect one-dimensional Weyl points at the steps. Since the local density of states is large at the step the system is susceptible to different types of instabilities, and we consider an easy-axis magnetization as one realistic possibility. We show that magnetic domain walls support a pair of zero-energy states because the regions with opposite magnetization are topologically distinct in the presence of non-symmorphic chiral and mirror symmetries, providing a possible explanation for the zero-bias conductance peak observed in the recent experiment [Mazur et al., arXiv:1709.04000].